Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #89825⋊C3 |
| Fitting | C7×#123 ⋊ C2×C3 |
| Hall | #89825 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #458 |
| Sylow | C7 ⋊ C3×#458 |
320 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^3,3,2^3,7|c^2=g,c^a=cf,e^a=eg,h^a=h^6,c^b=ce,f^b=fg,h^d=h^262 conjugacy classes — view the full character table.
| maximal subgroup of | #72277002, AGL(1,𝔽7)×#458, #72277159, … |
| maximal normal subgroup of | #72277002, AGL(1,𝔽7)×#458, #72277159, … |
| maximal quotient of | #72277260 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1242);